A pentagon of identities, graded tensor products and the Kirillov-Reshetikhin conjecture

This paper provides a brief review of the relations between the Feigin-Loktev conjecture on the dimension of graded tensor products of \g[t]-modules, the Kirillov-Reshetikhin conjecture, the combinatorial ``M=N" conjecture, their proofs for all simple Lie algebras, and a pentagon of identities which results from the proof.Comment: 21 page

Modular values and weak values of quantum observables

The concept of a modular value of an observable of a pre- and post-selected quantum system is introduced. It is similar in form and in some cases has a close connection to the weak value of an observable, but instead of describing an effective interaction when the coupling is weak, it describes a coupling of any strength but only to qubit meters. The generalization of the concept for a coupling of a composite system to a multi-qubit meter provides an explanation of some current experiments.Comment: 4 page